Initial distribution function of the electron and the positive hole in liquid cyclohexane determined by the geminate recombination data and the smoluchowski equation

1984 ◽  
Vol 23 (1-2) ◽  
pp. 279-283 ◽  
Author(s):  
Y. Yoshida ◽  
S. Tagawa ◽  
Y. Tabata
Keyword(s):  
Distribution Function ◽  
Initial Distribution ◽  
Smoluchowski Equation ◽  
Geminate Recombination ◽  
Positive Hole ◽  
Initial Distribution Function

scholarly journals Analytical solution of integro-differential equations describing the process of intense boiling of a superheated liquid

2021 ◽  
Author(s):  
Irina Alexandrova ◽  
Alexander Ivanov ◽  
Dmitri Alexandrov
Keyword(s):  
Distribution Function ◽  
Analytical Solution ◽  
Approximate Analytical Solution ◽  
Initial Distribution ◽  
Size Distribution Function ◽  
Superheated Liquid ◽  
Balance Equations ◽  
The Laplace Transform ◽  
And Shifts ◽  
Initial Distribution Function

In this article, an approximate analytical solution of an integro-differential system of equations is constructed, which describes the process of intense boiling of a superheated liquid. The kinetic and balance equations for the bubble-size distribution function and liquid temperature are solved analytically using the Laplace transform and saddle-point methods with allowance for an arbitrary dependence of the bubble growth rate on temperature. The rate of bubble appearance therewith is considered in accordance with the Dering-Volmer and Frenkel-Zeldovich-Kagan nucleation theories. It is shown that the initial distribution function decreases with increasing the dimensionless size of bubbles and shifts to their greater values with time.

scholarly journals Boltzmann Equation Theory of Charged Particle Transport in Neutral Gases: Perturbation Treatment

1999 ◽  
Vol 52 (6) ◽  
pp. 999 ◽  
Author(s):  
Slobodan B. Vrhovac ◽  
Zoran Lj. Petrovic
Keyword(s):  
Distribution Function ◽  
Perturbation Theory ◽  
Charged Particle ◽  
Boltzmann Equation ◽  
Particle Transport ◽  
Kinetic Equations ◽  
Initial Distribution ◽  
Formal Structure ◽  
Time Dependent ◽  
Charged Particle Transport

This paper examines the formal structure of the Boltzmann equation (BE) theory of charged particle transport in neutral gases. The initial value problem of the BE is studied by using perturbation theory generalised to non-Hermitian operators. The method developed by R�sibois was generalised in order to be applied for the derivation of the transport coecients of swarms of charged particles in gases. We reveal which intrinsic properties of the operators occurring in the kinetic equation are sucient for the generalised diffusion equation (GDE) and the density gradient expansion to be valid. Explicit expressions for transport coecients from the (asymmetric) eigenvalue problem are also deduced. We demonstrate the equivalence between these microscopic expressions and the hierarchy of kinetic equations. The establishment of the hydrodynamic regime is further analysed by using the time-dependent perturbation theory. We prove that for times t ? τ0 (τ0 is the relaxation time), the one-particle distribution function of swarm particles can be transformed into hydrodynamic form. Introducing time-dependent transport coecients ? *(p) (?q,t), which can be related to various Fourier components of the initial distribution function, we also show that for the long-time limit all ? *(p) (?q,t) become time and ?q independent in the same characteristic time and achieve their hydrodynamic values.

scholarly journals MODELING OF MULTICHANNEL QUEUING SYSTEMS AS COMPONENTS OF SOCIOTECHNICAL SYSTEMS

2018 ◽  
pp. 109-116
Author(s):  
Georgiy Aleksandrovich Popov
Keyword(s):  
Distribution Function ◽  
Recurrence Relations ◽  
Initial Distribution ◽  
Individual Characteristics ◽  
Distribution Functions ◽  
Technical System ◽  
Switching Function ◽  
Practical Implementation ◽  
Conflict Situations ◽  
Gamma Distributions

For a multichannel queuing system, in which all calls have individual characteristics of arrival and maintenance in accordance with their characteristics in the required socio-technical system and switching from one call to another are performed in accordance with a specified switching function determined by the resolution policy adopted in the socio-technical system of conflict situations, recurrence relations have been obtained for the queue lengths, for the list of free instrument changes, for the list of call numbers waiting for service, and for a number of other characteristics at successive call termination services. The procedure of sequential calculation of all specified characteristics of the system is described taking into account their internal interrelation. In accordance with this procedure, in a strictly defined sequence, eleven characteristics of the system are calculated on the basis of recurrence relations. To increase the efficiency of the process of practical implementation of the modeling procedure, it is proposed to replace the initial distribution functions of random variables by their approximate values, which are mixtures of gamma distributions whose values can be calculated significantly faster than the values of the initial distributions. The problem of finding a set of exponential distributions for a given simulated distribution function is formalized, the mixture of which approximates the distribution function with a given accuracy.

scholarly journals Clayton Copula as an Alternative Perspective of Multi-Reaction Model

2018 ◽  
Vol 22 (1) ◽  
pp. 83-106 ◽  
Author(s):  
Alok Dhaundiyal ◽  
Suraj B. Singh ◽  
Muammel M. Hanon ◽  
Norbert Schrempf
Keyword(s):  
Activation Energy ◽  
Distribution Function ◽  
Numerical Solutions ◽  
Analytical Data ◽  
Initial Distribution ◽  
Reaction Model ◽  
Bivariate Distribution ◽  
Clayton Copula ◽  
Scale Parameters ◽  
Temperature Ramp

Abstract This study proposes to assess the effect of some relevant parameters of biomass pyrolysis on the numerical solutions of nthorder distributed activation energy model (DAEM) or multi reaction model (MRM). The two-step process mechanisms of pyrolysis is described by replacing the initial distribution function of f (E) with the Clayton copula. The upper limit (E∞) of ‘dE’ integral, activation energy (A), heating rate (m), and the shape and scale parameters of bivariate distribution function. Temperature ramp rate is assumed to vary linearly with time. Thermo-analytical data is obtained with the help of thermogravimetric (TG) analysis. Asymptotic technique is adopted to approximate double exponential and bivariate distribution function f (E1, E2), where E1and E2are the activation energies for bivariate scheme.

scholarly journals The Non-Linear Theory of Spiral Structure

1977 ◽  
Vol 45 ◽  
pp. 229-240
Author(s):  
G. Contopoulos
Keyword(s):  
Distribution Function ◽  
Linear Theory ◽  
Periodic Orbits ◽  
Spiral Structure ◽  
Initial Distribution ◽  
Integral Of Motion ◽  
Density Maximum ◽  
Non Linear ◽  
Consistent Solution ◽  
Quadrupole Term

AbstractThe main steps of the non-linear theory of spiral structure are described. Near each of the main resonances the basic periodic orbits are calculated, and the sets of non-periodic orbits that follow them are found. A different integral of motion is applicable for each set, besides the Jacobi integral. Then the initial distribution function, f, is expressed as a function of the two integrals and the corresponding angles. The final distribution function is found by averaging over the angles:Then by integratingover all velocities we find the response density σresp. In order that σrespshould be equal to the imposed density, σimpwe must adjust the parameters of the imposed spiral field. The form of σrespaway from resonances can be derived explicitely for tight ana open spirals or bars; however near the resonances σrespcan be only calculated numerically. If the imposed field has almost constant amplitude, then the amplitude of the response is very large near the Inner Lindblad Resonance. In the case of a tight spiral the azimuth of the response density maximum with respect to the imposed density maximum tends to zero outside the ILR, while it tends to -90° inside the ILR. One possible self-consistent solution has zero amplitude inside the ILR both in the case of tight spirals and of bars. Finally an important quadrupole term was found near the ILR.

scholarly journals Statistical description of coalescing magnetic islands via magnetic reconnection

2021 ◽  
Vol 87 (6) ◽  
Author(s):  
Muni Zhou ◽  
David H. Wu ◽  
Nuno F. Loureiro ◽  
Dmitri A. Uzdensky
Keyword(s):  
Distribution Function ◽  
Kinetic Equation ◽  
Magnetic Reconnection ◽  
Magnetic Energy ◽  
Initial Distribution ◽  
Two Dimensions ◽  
Magnetic Islands ◽  
Plasma Dynamics ◽  
Dirac Delta ◽  
Physical Environments

The physical picture of interacting magnetic islands provides a useful paradigm for certain plasma dynamics in a variety of physical environments, such as the solar corona, the heliosheath and the Earth's magnetosphere. In this work, we derive an island kinetic equation to describe the evolution of the island distribution function (in area and in flux of islands) subject to a collisional integral designed to account for the role of magnetic reconnection during island mergers. This equation is used to study the inverse transfer of magnetic energy through the coalescence of magnetic islands in two dimensions. We solve our island kinetic equation numerically for three different types of initial distribution: Dirac delta, Gaussian and power-law distributions. The time evolution of several key quantities is found to agree well with our analytical predictions: magnetic energy decays as $\tilde {t}^{-1}$ , the number of islands decreases as $\tilde {t}^{-1}$ and the averaged area of islands grows as $\tilde {t}$ , where $\tilde {t}$ is the time normalised to the characteristic reconnection time scale of islands. General properties of the distribution function and the magnetic energy spectrum are also studied. Finally, we discuss the underlying connection of our island-merger models to the (self-similar) decay of magnetohydrodynamic turbulence.

Sign in / Sign up

Export Citation Format

Share Document